The present invention relates to arithmetic processing and, more particularly, to novel means, utilizing a plurality of COordinate Rotation Digital Computer complex multipliers, for performing the cross-correlation of a pair of complex sampled signals.
Signal processing functions of great relative complexity are utilized in many forms of modern electronics equipment. One such function is the cross correlation function of two complex sampled signals, evaluated over S sampled time intervals, so that: ##EQU1## where C, X and Y are all complex digital data signals with independent real and imaginary parts. Thus, a large number of multipliers and accumulators are normally required to cross-correlate a first complex digital data input signal X with a second digital data complex signal Y, over S sampling periods, to yield a plurality M of complex digital data output signals C(m), C(m+1), . . . , C(m+M-1), where M is a total number of time delays at which the correlation function is evaluated. It is highly desirable to replace all multipliers with circuitry realizing the multiplication function with utilization of more-integratable building blocks, such as adders, shift registers and the like.
Several forms of complex multipliers, utilizing CORDIC apparatus and techniques, are described and claimed in our copending allowed U.S. application Ser. No. 200,491, filed May 31, 1988, which corresponds to U.S. Pat. No. 4,896,287, assigned to the assignee of the present invention and incorporated herein in its entirety by reference. It is highly desirable to provide complex cross-correlation apparatus utilizing CORDIC complex rotators wherever possible, especially as such a cross-correlation apparatus can be fully integrated into a single semiconductor circuit chip. The CORDIC system allows rotation through an angle .theta. to be represented as the summation of several sequential rotations, with each rotation being through one of a special set of angles .alpha., such that ##EQU2## where .xi..sub.i =+1 or -1. If a first angle .alpha..sub.1 =90.degree. is defined, then subsequent angles .alpha. (for a maximum of N rotations) are given by EQU .alpha..sub.n+2 =tan.sup.-1 (2.sup.-n), n=0, 1, 2, . . . , N-2
so that the total angle is successively approximated using all of the plurality n of angles .alpha..sub.i, with each finer approximation of the angle .theta. providing rectangular-coordinate results x.sub.n+1 and y.sub.n+1, which are related to a pair of x.sub.n and y.sub.n rectangular-coordinate values for the next-coarsest approximation by the pair of equations: EQU x.sub.n+1 =K(.theta..sub.i)(x.sub.n +.xi..sub.i y.sub.n /2.sup.n) EQU y.sub.n+1 =K(.theta..sub.i)(y.sub.n -.xi..sub.i x.sub.n /2.sup.n),
where K(.theta..sub.i) is a scale factor equal to cos (.theta.). Since each of the 2.sup.-n factors is in effect a division-by-two operation done n times, and is provided, for binary numbers, by a one-bit shift for each of the n occurrences, the complex multiplication can, except for the scale factor K(.theta..sub.i) multiplication (if needed), be carried out with a set of inverters, multiplexers, registers and adders. It is this technique that is desirable to be applied to a cross-correlator for a pair of complex digital signals.